Painless Trigonometry: a tool-complementary
school mathematics project
Jesús Jiménez-Molotla, [email protected]
Escuela Secundaria Diurna #229 “Ludmila Yivkova”, Turno Matutino, DF, Mexico
Alessio Gutiérrez-Gómez, [email protected]
Escuela Secundaria Diurna #229 “Ludmila Yivkova”, Turno Matutino, DF, Mexico
Ana Isabel Sacristán, [email protected]
Dept. of Mathematics Education, Centre for Research & Advanced Studies (Cinvestav), Mexico
Abstract
Since 2001-02, we have been working in our mathematics classrooms with the materials and
digital tools provided by a government-sponsored national programme: the Teaching
Mathematics with Technology programme (EMAT). The main computer tools of the EMAT
programme are Spreadsheets (Excel), Dynamic Geometry (Cabri-Géomètre), and Logo
(MSWLogo). At the beginning we used these tools independently, but in more recent years we
have tried to develop long-term projects that incorporate all the tools, and that also serve as
means to introduce students to topics of mathematics that are normally considered too
advanced for them (such as trigonometry).
In this paper we report the design and results of one of those projects, a trigonometry project,
that we first implemented in the academic year 2005-06 with 6 groups of the first two grades of
two junior secondary schools in Mexico. Approximately 250 students of 12-14 yrs of age
participated, in total, in the project in that year.
We believe in the importance of using, in an integrated and complementary way, a variety of
tools for learning since we consider that each tool brings with it a different type of knowledge and
constitutes a different epistemological domain (Balacheff & Sutherland, 1994). We also believe
in the importance of constructionist (Harel & Papert, 1991) or programming activities for
meaningful learning. With those fundamental theoretical premises, we developed the long-term
trigonometry project for introducing young students to the Pythagorean theorem, basic
trigonometry concepts, and their applications using explorations and constructive activities with
Cabri-Géomètre, Excel and Logo (Figure 1). Students thoroughly enjoyed the activities and
gained interest in mathematics. They also developed problem-solving and collaborative skills.
Furthermore, in written tests after the project, the students showed an understanding of the
“advanced” trigonometry concepts, as well as of other algebraic ideas.
Figure 1. Complementary trigonometry explorations and constructions with Cabri, Excel and Logo
Keywords
Mathematics; trigonometry; Pythagorean theorem; Logo; Excel; Cabri-Géomètre; school project
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Jesús Jiménez-Molotla, Alessio Gutiérrez-Gómez, Ana Isabel Sacristán
…children can learn to use computers in a masterful way,
and […] learning to use computers can change the way they
learn everything else… (Papert, 1980; p. 8)
Background
In 2001-2002, the Teaching Mathematics with Technology (EMAT) Programme began to be
implemented in the junior secondary schools we work at, in Mexico City, Mexico. EMAT is a
programme sponsored, since 1997, by the Mexican Ministry of Education to promote the use of
new technologies, using a constructivist approach, to enrich and improve the current teaching
and learning of junior secondary mathematics in Mexico. A study carried out in Mexico and
England (Rojano et al., 1996) involving mathematical practices in science classes, revealed that
in Mexico few students were able to close the gap between the formal treatment of the curricular
topics and their possible applications. This suggested that it was necessary to replace the formal
approach of the then official curriculum, with a “down-up” approach capable of fostering the
students’ explorative, manipulative, and communication skills. Thus, a main part of the EMAT
programme are pre-designed activities and a pedagogical model that emphasize exploratory and
collaborative learning. And the various EMAT software were chosen on the criteria (Ursini &
Rojano, 2000) that they be open tools; that is, where the user could be in control and have the
power of deciding how to use the software. These open tools had to be flexible enough so that
they could be used with different didactical objectives (those of the pre-designed activities, or
others as well). The main computer tools of the EMAT programme are Spreadsheets (Excel),
Dynamic Geometry (Cabri-Géomètre), and Logo (MSWLogo).
Initially at our schools, we worked with each of these tools separately, covering different themes
with each of them. Logo was the last tool that we incorporated into our schools, and one of the
immediate things we noticed was how much it enriched, not only children’s motivations for
exploring mathematical topics, but also the use of the other tools: because of the programming
experience with Logo, students began asking if it was possible to also program the other tools
(this led us, for example, to show them how to create macros in Excel and Cabri).
Moreover, we began to look into ways of integrating the three tools. Each tool brings with it a
different type of knowledge and constitutes a different epistemological domain (Balacheff &
Sutherland, 1994). This, in fact, is one of the reasons why the EMAT designers provided various
tools, but their use has generally been independent one from the other. We consider it, however,
important to use these tools in an integrated and complementary way.
The trigonometry project
We thus took up the challenge, in the academic year 2005-06, to create an ICT-based project for
the learning of a topic which is traditionally difficult to teach and learn, and more so at the junior
secondary level (children aged 12 to 15 yrs-old): trigonometry. This is a topic which is in the
curriculum only for Grade 3 (the older students) of the junior secondary level; but we decided to
work with the younger students of Grades 1 and 2. On the one hand, we concur with the idea
that ICT can act as scaffolding for learning more advanced topics; on the other, our strategy was
to familiarize students with this topic so that by the time they get to Grade 3 and have to formally
learn trigonometry, they will have experiences and useful intuitive ideas (diSessa, 2000) to build
upon. For Grade 1 students, we would build upon the curricular themes of operations with
decimal numbers, square root and powers, and from there create a link to the trigonometry
project. In the case of students of Grade 2 we would have less difficulty, as these children
already work with algebra; we could thus link it to curricular themes of polynomials and the use
of variables as well as those of square root, geometrical figures.
Our approach was to use not only the EMAT tools –Logo, Cabri and Excel— but also
Powerpoint presentations and paper-and-pencil activities, to introduce the theme and the use
(for this project) of the various tools.
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Painless Trigonometry: a tool-complementary school mathematics project
The stages and implementation of the project
One of the first aims was to combat the apathy that the majority of Grade 1 students have
towards mathematics. The use of the technological tools provided a means to motivate our
students.
In the first sessions of the school-year we taught students, of both grades, the basic use of the
three tools (Logo, Cabri and Excel) through some of the pre-designed EMAT activities (students
of neither group had used any of these tools before –although in our school the tools have been
available for a few years, the teacher that the Grade 2 students had had the previous year, did
not make use of them).
Later, we began to engage students in the trigonometry project explorations: For this project, we
began with an integrated theoretical-practical Powerpoint presentation and a paper-and-pencil
exercise, followed in later sessions, with technology-based explorations, as described further
below.
The explorations were carried over at least a dozen weekly technology-based sessions (i.e. one
50 minute session, once a week). On the one hand, the long-term investigation over the course
of the year was motivating and engaging for everyone; it also provided time between sessions to
rethink and re-orchestrate the strategies, if necessary. On the other hand, we had the problem,
for students, of the long weekly (and sometimes longer) gap between sessions; also, in the case
of the group that had their weekly technology sessions, at the end of the week, we noticed
weariness that slightly affected the sessions.
The presentation and paper-and-pencil exercise
After covering the curricular theme “Square Roots with Decimals”, we began with a Powerpoint
presentation and an exercise on the Pythagorean theorem. In the exercise a square is drawn
inside a 7 X 7 grid in such a way that four right-angle triangles are formed (each with legs of
sizes 3 and 4), with the sides of the square being the hypotenuses (see Figure 2). In this
example, the length of the hypotenuse is the integer 5, but we clarified that it was an exception.
The aim of this exercise was to illustrate (see Figure 3) that the sum of the squares of the legs of
each right triangle was the same as the square of the length of the hypotenuse (the area of the
inside square).
Figure 2. The first step in the Pythagorean
Theorem exercise
Figure 3. The last slide in the Pythagorean
Theorem presentation
The technology-based explorations
The previous exercise served simply as an introduction to the Pythagorean theorem and as
support for the software-based explorations that followed, where students would investigate
3
Jesús Jiménez-Molotla, Alessio Gutiérrez-Gómez, Ana Isabel Sacristán
further this theorem and its application through construction activities with Cabri, Excel and
Logo. In these technology-based activities, the students were the main actors, collaborating and
directing the construction process and their peers. As recommended by the EMAT pedagogical
model, students worked in teams of two (or sometimes three), with one computer. The
structuring of teams was important for discussing and collaborating on the solutions to the
problems, and confronting different strategies. The teams then had the opportunity to present to
the entire group their work, and engage in whole-classroom discussions (which is also an
important recommendation of the EMAT pedagogical model).
(It may be worth mentioning that we did have a difficulty towards the end of the year, when some
of our projection equipment was damaged and whole-group presentations were not as easy; but
this is part of the challenges of working with technology.)
The Cabri and Excel explorations
The first technology-based activities, were geometric explorations of the Pythagorean theorem
using Cabri (see Figure 4), to illustrate that the size of the hypotenuse of a right triangle can
always be determined by the sizes of the legs of the triangle; we also introduced the
trigonometric circle and basic trigonometric concepts such as sine, cosine and tangent.
With Excel, we introduced exponential formulas for powers and square roots, as well as
trigonometric formulas which were related to the explorations with Cabri. In this Excel
exploration (see Figure 5), the students programmed the Pythagorean theorem formulas for
finding the values of the legs and hypotenuse of a right triangle; as well as formulas, using
trigonometric functions, for finding the value of missing angles. They also programmed
interactive buttons, something which was motivating for them, and that forced them to
understand better the underlying relationships.
Figure 4. The trigonometry explorations with Cabri.
Figure 5. The Excel explorations of the Pythagorean
theorem and of trigonometric functions
The Logo explorations
With Logo, we divided the explorations in several stages:
In the first stage, students had to construct a specific case: a right triangle with legs 30 and 40
(which meant the hypotenuse was a nice, and easy to compute, integer). They soon learned (out
of need) to use the primitives arcsin, arctan, and arcos, since without them they would not have
been able to compute the rotation angles for drawing the triangle. In this sense, the previous
activities with Cabri and Excel were very useful; the first, because it gave them an understanding
of the underlying concepts of the trigonometric functions; and the latter, because it familiarized
them with the formulas and use of the trigonometric functions. Now, in Logo, in order to use the
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Painless Trigonometry: a tool-complementary school mathematics project
formulas for finding the missing angles, they really needed to be able to read them properly, and
to understand exactly how to apply them in the new context (a situation where, in many cases –
depending on how the procedure was constructed—, the angles needed were the rotation
angles of the triangle, instead of the inner ones).
In the second stage, the task was to write programs for other right triangles, using any sides
(legs) of the triangle they wanted. This time (if they hadn’t done it before) they had to use, within
their procedure, the formula derived from the Pythagorean theorem and the sqrt primitive
command for the square root, in order to generate the hypotenuse. At a first attempt, they could
compute with paper-and-pencil, or in Logo’s direct mode, the rotation angles and simply use the
values in their procedure (Procedure 1). But they later had to also program, within the
procedure, the formula for the rotation angle (see Procedure 2).
Procedure 1
to pitagoras
fd 100
bk 100
rt 90
fd 100
lt 135
fd sqrt(100 * 100 + 100 * 100)
end
Procedure 2
to pitagoras2
fd 200
bk 200
rt 90
fd 150
lt 180
rt arctan (200 / 150)
fd sqrt (200 * 200 + 150 * 150)
end
The latter was the stepping-stone for the final generalization stage, in which students had to
write a general program for a right triangle, using variables. Furthermore, in this stage, students
not only had to construct the procedure for the right triangle, but at the end, they were also
asked to write a procedure that would draw the squares on each side of the triangle and
compute their values for labelling the figure. In this way, they built a model of the concept of the
Pythagorean theorem (see Procedure 3 and Figure 6).
Procedure 3: Final procedure for the Pythagorean model programmed by one of the teams of students
to triangulo :x :y
fd :x bk :x rt 90
setcolor 20
fd :y lt 180
wait 30 rt arctan :x / :y
wait 30 fd sqrt(:x * :x + :y * :y)
wait 30 setcolor 10 rt 90 repeat 3 [fd sqrt(:x * :x + :y * :y) rt 90]
lt 90 + arctan :x / :y
wait 30 setcolor 5 repeat 4 [fd :y rt 90]
rt 90
fd :y
wait 30 setcolor 20 repeat 4 [fd :x rt 90]
rt 90
fd :x rt arctan :x / :y
label sqrt (:x * :x + :y * :y)
end
When students had completed their programs, we let them play around freely with their
procedures (see Figure 7). This type of activities are important to relax, motivate and reward the
students for their hard work.
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Jesús Jiménez-Molotla, Alessio Gutiérrez-Gómez, Ana Isabel Sacristán
Figure 6. The Pythagorean model produced by the
final procedure in Logo.
Figure 7. Example of some students playing with
their right triangle procedures in Logo.
Some results
In other papers (Sacristán, 2003; Sacristán et al., 2006) some of the benefits (as well as
difficulties) of the implementation of the EMAT programme, its tools and pedagogical model,
have already been reported, in particular those concerning the development of skills and
motivation in students. Here we present the particular observations from our schools.
Observations on the development of problem-solving abilities and collaborative attitudes
Since we began integrating the EMAT ICT tools for our mathematics courses, and particularly
Logo and other “programming” or constructionist (Harel & Papert, 1991) activities, we have
noticed that our students have developed many valuable learning skills: In particular, there is an
increase in their problem-solving abilities, and they tend to reflect more on the problems and
even go beyond the tasks required.
In fact, both the students and the teachers involved in the EMAT programme at our schools and
in the project presented in this paper, have found new means of solving problems or doing
things. Because we do not impose a particular way for solving the tasks, there is more freedom
to analyse, conjecture, test, discuss, compare and learn from each other, with a collaboration
between the students, the math teacher and the ICT teacher.
We also believe that the team-work during the sessions, and the sharing and discussing of
solutions and strategies at the end of each session, are fundamental elements for reaching
successful solutions and for stabilizing the learning derived from the technology-based activities
and explorations.
Achievements in School Assessments
In the end-of-term mathematics tests, the students involved in our courses had very high scores,
with a high percentage of success rate (see Table 1) and it is worth noting that parts of these
tests covered trigonometry and algebra, even for first grade students. Samples of solved tests
are given in Figure 8.
6
Painless Trigonometry: a tool-complementary school mathematics project
Group
1° A
1° B
1° C
1° D
Nb. of students with
pass grade
35
34
32
42
Nb. of students with
fail grade
9
10
15
5
Table 1. Results of the end-of-term mathematics tests for Grade 1
Figure 8: Written tests on Trigonometry by a Grade 1 student and a Grade 2 student, respectively
Observations on affective aspects
With the use of the ICT tools and activities in our mathematics courses, we have noticed a much
greater interest on the part of the students for mathematics. This increased when we
incorporated Logo, and it became even greater when we began to engage on long-term projects
such as the one presented in this paper.
In fact, we face very strong difficulties in making students leave the technology-based sessions,
because they are so involved and enjoy their work there so much. When they do leave, they go
out discussing amongst themselves their different or possible solutions and procedures.
The atmosphere and dynamics of the classrooms have also changed deeply. We have observed
much improved classroom discipline and engagement, with a reduction of behavioural problems.
Also students tend to get along better and they have developed values of cooperation and
assistance amongst themselves: they appreciate the benefits of collaborative work.
Most importantly, through these technology-based activities, students are given the opportunity
to find their own paths for reaching a solution. When they achieve this, they gain confidence in
their own knowledge and abilities and this is highly motivating for them.
Remark on students’ perceptions of the tools
In terms of the technological tools, the involvement in the project (as well as with other EMAT
activities) have shown students the various uses they can give to tools like Logo, Cabri and
Excel and how each of these tools complements the other. They have become proficient in the
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Jesús Jiménez-Molotla, Alessio Gutiérrez-Gómez, Ana Isabel Sacristán
use of all these tools. Furthermore, students no longer make a memorized “algorithmic” use of
the tools, but they actually reflect on what they are doing, why, and how.
Final remarks
What we have achieved with our projects, is to make mathematics explorations, an enjoyable
part of students’ life, where working towards finding a solution actually becomes a game for
them –like it does for a real mathematician. We observed how they used names for their Logo
procedures and objects that perhaps didn’t reflect what they were doing –we let them, because
these objects were their toys, and the explorations became their games. And this is what we
believe motivates learning.
Our students our now able to perceive mathematics as something meaningful, something that
they can apply for solving a project; instead of something boring, meaningless and forced upon
them –as is so often the case. Furthermore, they can use the technological tools as aids for
developing their abilities, and learning how to think, instead of simply “clicking buttons”.
From our side, we also achieved to introduce what is normally considered a "too advanced" topic
for Grade 1 students, and showed (as is seen in the sample tests in Figure 8) that children did
learn and understand some of the ideas they explored: not only the trigonometry ideas, but also
the use of algebraic variables and concepts. In its 40 years, Logo has been proved an excellent
tool for introducing algebraic concepts and particularly the use of variables. In our project, we
have again confirmed that. Whatever may happen, we are now at ease knowing that most of our
students are proficient in the curricular topics of Grades 1 and 2, and beyond. Richard Noss told
us once, that “knowledge has no age; it rather depends on how the teacher wants to teach it”
(personal communication).
We feel that every mathematics programme should have some programming (or at least,
constructionist) activities; in our case it was Logo; and its influence made us develop more
constructive activities with the other tools. We also appreciate the conceptual perspectives that
each of the tools we used, offered. Cabri, with its manipulable dynamic representations allowed
us to explore properties of right triangles, such as the relationship between the sides, and
between the sides and the angles; it also gave us a means to introduce the trigonometric circle
and a basic understanding of some trigonometric functions. With Excel (and Logo), we delved
deeper into the structure of the Pythagorean and basic trigonometric formulas and of the
algebraic relationships between their elements. With Logo, we gained a further understanding of
how and when to apply those concepts for solving a particular project: that of drawing a general
right triangle. It was an orchestration of all the tools.
But it was not easy. We faced many challenges –e.g. from difficulties in using the computer labs,
to learning how to adapt how we teach. It has taken us several years of experience working with
technology in our mathematics classrooms for us to gradually change our practice, the way we
teach, and the ways we perceive and use the tools. As is reported in Sacristán et al. (2006), one
of the biggest difficulties in the implementation of the EMAT programme, has been teachers’
difficulties in adapting to the proposed pedagogical model and changing their classrooms’
dynamics. We also had those difficulties. A few years ago, we were not used to having children
talk and discuss during class. Now, that has changed. But we have to be open to the possibility
of change, before that can happen, and we understand how difficult that can be for many of our
peers.
In fact, we faced many criticisms (and even obstacles) from some peers and authorities, when
we began to develop our technology-based projects, because they didn’t understand what we
were doing, despite the fact that the EMAT programme is sponsored by the government.
Again, change is hard. Many of our peers fear the use of technology because they don’t feel
proficient enough (as is also reported in Sacristán, 2006); they also fear changing the way they
were used to teach. We also faced those fears, but despite our insecurities, we took the risk and
have enjoyed discovering we can learn at the same time as the students. Much later we
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Painless Trigonometry: a tool-complementary school mathematics project
discovered that, in fact, Papert (1999, p. xv) believes “that there is such a thing as becoming a
good learner and therefore that teachers should do a lot of learning in the presence of the
children and in collaboration with them”.
Another fear that teachers have, is in engaging in such long-term projects because they feel they
loose time that they need to cover the required curricular topics. What we have found out, is that
many of the curricular topics naturally arise from the explorations. Projects, such as the one
described in this paper, not only deal with the topic they are designed for; the need for the use of
many other mathematical concepts also emerges 1 and we are reminded of Papert’s vision in his
book Mindstorms when he described the gears of his childhood (Papert, 1980). This gives us the
opportunity to introduce new mathematical topics as children encounter them, and then we can
develop them. In this way, the students have a means to relate to the mathematics, and it
becomes more meaningful for them. Thus, time is far from being lost; it is actually well invested
in activities that are much more significant for students than the usual rushing through all the
curricular topics.
We believe that in the future, if students learn to use technology in a thoughtful way (moving way
from a mechanized use) for understanding, developing projects or solving problems they will
become better learners and the teacher will have more of mediating role between students and
mathematics rather than being the traditional presenter of knowledge.
We are a new generation of students and teachers creating experimental environments,
computational microworlds and classroom dynamics that are very different from traditional
school practices. The catalyst for change are the technological tools, but it is in our power, as
teachers, to actually make the change, and use the tools in innovative ways that change
mathematical teaching and learning. The tools don’t bring the benefits; it’s the use we make of
them that does.
In the words of Seymour Papert (1999, p. xv):
Opportunity means more than just “access” to computers. It means an
intellectual culture in which individual projects are encouraged and contact with
powerful ideas is facilitated.
Doing that means teachers have a harder job. But we believe that it is a far
more interesting and creative job and we have confidence that most teachers
will prefer “creative” to “easy.”
In that spirit, we continue working, as we have for the past 5 years, in developing interesting
mathematical projects for our students with an integral use of technological tools like Logo, Cabri
and Excel. And we hope that by sharing our experience, others will also dare change, use
technologies in meaningful ways and perhaps also create their own projects, where there is
collaboration and partaking of experiences amongst students, between students and teachers,
and also amongst teachers, that enriches everyone’s learning.
And if a student of any age comes to us for learning trigonometry, or any other topic, we hope to
be able to give him/her the means to learn it “painlessly”.
References
Balacheff, N. and Sutherland, R. (1994) Epistemological Domain of Validity of Microworlds: The
Case of Logo and Cabri-géomètre. In Lessons from Learning, IFIP Conference TC3WG3.3, R.
Lewis & P. Mendelsohn (eds), North Holland, pp. 137-150.
1
In the project described here, as explained before, the students not only explored the basic
trigonometric ideas, but it also allowed us to cover many curricular topics and beyond, from
number operations to algebraic ideas, such as variables, use of formulas and manipulation of
equations.
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Jesús Jiménez-Molotla, Alessio Gutiérrez-Gómez, Ana Isabel Sacristán
diSessa, A. A. (2000). Changing minds: Computers, learning, and literacy. Cambridge: MIT
Press.
Harel, I., & Papert, S., (eds.) (1991). Constructionism. Norwood: Ablex.
Papert, S. (1980). Mindstorms: Children, Computers, and Powerful Ideas, New York: Basic
Books.
Papert, S. (1999) What is Logo? Who Needs It? In Logo Philosophy and Implementation, Logo
Computer Systems Inc. Pp. iv-xvi.
Rojano, T., Sutherland, R., Ursini, S., Molyneux, S., and Jinich, E. (1996). Ways of solving
algebra problems: The influence of school culture. In Proceedings of the 20th Conference of the
International Group for the Psychology of Mathematics Education, Vol. 4. L. Puig & A. Gutierrez
(Eds.), Valencia, Spain: Universidad de Valencia. pp. 219-226.
Sacristán, A. I. (2003) Mathematical Learning with Logo in Mexican Schools. In Eurologo’2003
Proceedings: Re-inventing technology on education. Coimbra, Portugal: Cnotinfor, Lda., pp.
230-240.
Sacristán, A. I., Ursini, S., Trigueros, M. and Gil, N. (2006). Computational Technologies in
Mexican Classrooms: The Challenge of Changing a School Culture. In Information Technologies
at Schools. The 2nd International Conference “Informatics in Secondary Schools: Evolution and
perspectives” Selected papers, V. Dagiene and R. Mittermeir (Eds.). Vilnius: Institute of
Mathematics and Informatics. Pp. 95-110.
Ursini, S. & Rojano, T. (2000). Guía para Integrar los Talleres de Capacitación EMAT. Mexico:
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Acknowledgements
We are grateful for the support of our schools’ authorities and of the following organizations in
Mexico: SEP, ILCE (the EFIT-EMAT team), OEI, Conacyt (with the research grants: No. G26338S
and No. 44632); as well as the encouragement of Professors Celia Hoyles and Richard Noss of
the University of London, UK.
We also thank Seymour Papert for his inspiration and dedicate this paper to him.
10